Robust Smith Predictor Design for
Time-Delay Systems with
H
∞Performance
Vinicius de Oliveira1 and Alireza Karimi2 Automatic Control Laboratory, Ecole Polytechnique F´ed´erale de
Lausanne (EPFL), Switzerland
Abstract: A new method for robust fixed-order H∞ controller design for uncertain time-delay
systems is presented. It is shown that theH∞robust performance condition can be represented
by a set of convex constraints with respect to the parameters of a linearly parameterized primary controller in the Smith predictor structure. Systems with uncertain dead-time and with multimodel and frequency-domain uncertainty can be considered straightforwardly in the proposed approach. Furthermore, it is shown that the design method can also be extended to design of robust gain-scheduled dead-time compensators. The performance of the method is illustrated by simulation examples.
1. INTRODUCTION
Most industrial processes present dead time in their dy-namics. Generally, dead times are caused by the time needed to transport energy, mass or information, but they also can be caused by processing time or by accumulation of time lags in a sequence of simple dynamic systems inter-connected in series (Normey-Rico and Camacho [2007]). The presence of dead times in the control loops has two main consequences: it greatly complicates the analysis and the design of feedback controllers and it makes satisfac-tory control performance more difficult to achieve (Palmor [1996]). Dead-time compensators can be used to improve the closed-loop performance of classical controllers (PI or PID controllers) for processes with delay. The Smith Predictor (SP) (See Fig.1), proposed in the late 1950s by Smith [1957], was the first dead-time compensation structure used to improve the performance of the classical controllers and became the most known and used algo-rithm to compensate dead time in the industry.
Although the SP offers potential improvement of the closed-loop performance of process with large dead-time, it requires a good model since small modeling errors can lead to very poor performance. In fact, even if the SP is nominally stabilizing, the closed-loop system may become unstable for infinitesimal change in the process dynamics (Palmor [1980]). For this reason, research efforts have been focused on robustness issues of the SP. A tuning method for models with one uncertain parameter is proposed by Brosilow [1979]. Easy tuning rules for SP in the presence of dead-time uncertainty is addressed in Santacesaria and Scattolini [1993] and a guideline for selection the closed-loop bandwidth based on the dead-time uncertainty bound is proposed. In Palmor and Blau [1994], a robust tuning rule is developed which considers the modeling error in the dead time. Robust PID tuning for SP considering model
1 Vinicius de Oliveira is currently a PhD student student at the
Department of Chemical Engineering of the Norwegian University of Science and Technology, Norway
2 Corresponding author:[email protected]
uncertainty is proposed by Lee et al. [1999]. In particular, first and second order plus dead-time systems which may contain uncertainty in multiple parameters of the model are considered. In Meinsma and Zwart [2000], tuning guidelines are presented for setpoint tracking considering model mismatches in the dead-time.
Recently, many researchers are interested in the optimal
control of dead-time systems, especiallyH∞ control, i.e.,
to find a controller to internally stabilize the system
and to minimize the H∞-norm of an associated transfer
function. Many relevant results have been presented in this framework using modified versions of the SP. See, for instance, Mirkin [2003], Meinsma and Zwart [2000] and Zhong [2003a]. Nonetheless, as pointed out by Zhong [2003b], these controllers are too involved and the predic-tor always includes additional unstable hidden modes even for stable plants. As the author argues, these hidden modes are not safe as they tend to destabilize the system when implemented. Therefore, the practical significance of these results are limited.
This paper presents a new method to design fixed-order SP controllers that considers uncertainty simultaneously in the dead-time and in the rational part of the model. The
performance specification, like the standard H∞ control
problem, is a constraint on the infinity norm of the weighted sensitivity function and is represented by a set of convex constraints in the Nyquist diagram. A line search
on the upper boundγ of the infinity norm of the weighted
sensitivity function can be used in association to a convex feasibility problem to determine the primary controller parameters. The proposed method can be used for PID controllers, which are of great practical interest, as well as for higher order linearly parametrized controllers in discrete or continuous time. The main idea is introduced for LTI-SISO systems and will be extended to MIMO systems in future works.
It is important to point out that a method to design fixed-order controllers for standard control loops based on convex constraints in the Nyquist diagram has been firstly proposed in Karimi and Galdos [2010]. This paper uses
the same framework to design dead-time compensators for time-delay systems. The extension to MIMO systems will be based on the idea presented in Galdos et al. [2010b] for
designingH∞ decoupling MIMO controllers.
This paper is organized as follows: In Section II the class of models, controllers and the control objectives are defined. Section III introduces the control design methodology based on convex constraints in the Nyquist diagram. In Section IV the results are extended to unstable delay systems. Gain-scheduled SP is designed for time-delay systems in Section V. Finally the concluding remarks are given.
2. PROBLEM FORMULATION 2.1 Class of models
Consider the class of stable time-delay LTI-SISO systems with bounded infinity norm. It is assumed that the plant model can be represented by:
P (s) = G(s)e−τs (1)
where the time delayτ is unknown but belongs to a finite
set{τ1, τ2, . . . , τq} and the dead-time free part of the model
has unstructured multiplicative uncertainty described as: G(s) = Gn(s)[1 + ∆(s)W2(s)] (2)
where W2(s) is a known stable uncertainty filter, Gn(s)
the nominal dead-time free model and ∆(s) an unknown
stable transfer function with∆∞< 1. Therefore, we can
assume thatP (s) belongs to a set P of q models given by:
P={Pi(s) = G(s)e−τis; i = 1, . . . , q} (3)
2.2 Class of controllers
The SP control structure shown in Fig.1 is considered.
The nominal model P0(s) = Gn(s)e−τns with a fixed τn
belongs to {τ1, τ2, . . . , τq} is used for the implementation
of the controller. + + + + − − r C(s) u P (s) Gn(s) e−τns yp y
Fig. 1. Smith Predictor
The primary controllerC(s) is linearly parametrized by
C(s) = ρTφ(s)
(4) whereρT = [ρ
1, ρ2, . . . , ρnc] is annc dimensional vector of
the controller parameters and φT
(s) = [φ1(s), φ2(s), . . . , φnc(s)] (5) is a vector of basis functions withφi(s) transfer functions
with no RHP poles. For instance, a PID controller could be linearly parametrized by ρT = [Kp, Ki, Kd] , φT(s) = [1, 1 s, s 1 +Tfs ] ++ − r Ceq(s) u P (s) y
Fig. 2. Equivalent control diagram 2.3 Design specifications
The SP can be represented by a classical feedback loop (see Fig. 2) whereCeq(s) is the equivalent controller given
by
Ceq(s) = C(s)[1 + C(s)H(s)]−1 (6)
withH(s) = Gn(s) − P0(s) = Gn(s)(1 − e−τns).
Let the sensitivity function of the closed-loop system with thei-th plant model Pi(s) be defined as
Si(s) = 1 1 +Pi(s)Ceq(s) = 1 +C(s)H(s) 1 +C(s)[H(s) + Pi(s)] (7) and the complementary sensitivity function as
Ti(s) = P i(s)Ceq(s) 1 +Pi(s)Ceq(s) = C(s)Pi(s) 1 +C(s)[H(s) + Pi(s)] (8) A standard robust control problem is to design a controller
that satisfies W1Si∞ < 1 for a set of models where
W1(s) is the performance weighting filter. If the model is
described by unstructured multiplicative uncertainty, the necessary and sufficient condition for robust performance is given by (Doyle et al. [1992]):
|W1Si| + |W2Ti|∞< 1 for i = 1, . . . , q (9)
The goal of the proposed approach is to design the primary
controller C(s) in the SP structure to guarantee robust
performance of the closed-loop system. 3. PROPOSED METHOD
The robust performance condition (9) can be written as: |W1(jω)Si(jω)| + |W2(jω)Ti(jω)| < 1, ∀ω (10)
for i = 1, . . . , q. The dependency on frequency ω and s will be omitted for brevity but the dependency on the
controller parameterρ will be highlighted.
LetLi(ρ) be defined as Li(ρ) = C(ρ)(H + Pi). Note that
this transfer function is not equal to the open loop transfer functionLeqi(ρ) = PiCeq(ρ) of the equivalent control loop
in Fig. 2. The main properties ofLi(ρ) are:
(1) linearity with respect to the controller parameterρ;
(2) the closed loop poles are the roots of 1 +Li(ρ) = 0.
The first property is clear becauseC(ρ) is linearly
param-eterized. To show the second property, let C(ρ) = Nc
Dc,
Gn = Ng
Dg be defined. The equivalent controller can be
written as Ceq(ρ) = N cDg DcDg+NcNg(1− e−τns) (11) Then, we have: 1 +Leqi(s) = DcDg+NcNg(1− e−τns+e−τis) DcDg+NcNg(1− e−τns) (12)
Note that the numerator of 1 + Leqi(s) is a
quasi-polynomial and its zeros are the closed-loop poles of the system. On the other hand, we have
1 +Li(ρ) = D
cDg+NcNg(1− e−τns+e−τis)
DcDg
(13)
It is clear that 1 +Li(ρ) shares the same zeros with 1 +
Leqi(s). This property will be used to prove the closed loop stability.
3.1 Main Result
The main result of this section is given in the following theorem.
Theorem 1. Consider the set of models P in (3) with
multiplicative uncertainty filter W2(jω), then the linearly
parametrized controller in (4) in the SP structure guaran-tees closed-loop stability and satisfy the following robust performance condition: |W1Si| + |W2Ti|∞< 1 for i = 1, . . . , q (14) if W1(jω)[1 + C(jω, ρ)H(jω)]+ W2(jω)C(jω, ρ)Pi(jω)|1 +Ld(jω)| − Re{[1 + L∗ d(jω)][1 + Li(jω, ρ)]} < 0 ∀ω fori = 1, . . . , q (15)
where Ld(jω) is a strictly proper transfer function which
does not encircle the critical point and L∗d(jω) is its
complex conjugate.
Proof : Since the real part of a complex number is less than or equal to its magnitude, we have
Re{[1 + L∗
d][1 +Li(ρ)]} ≤ |[1 + L∗d][1 +Li(ρ)]| (16)
Then, using (15) and the fact that|1 + Ld| = |1 + L∗d|, one
obtains W1(1 +C(ρ)H)+W2C(ρ)Pi − |1 +Li(ρ)| < 0 ∀ω for i = 1, . . . , q (17) UsingLi(ρ) = C(ρ)(H + Pi) we have |W1(1 +C(ρ)H)| + |W2C(ρ)Pi| |1 + C(ρ)(H + Pi)| < 1 ∀ω for i = 1, . . . , q (18) that leads directly to (14). To prove that all closed-loop transfer functions are stable, consider (15) which gives:
Re{[1 + L∗
d(jω)][1 + Li(jω, ρ)]} > 0 ∀ω (19)
or, alternatively, wno{[1 + L∗
d(jω)][1 + Li(jω, ρ)]} = 0 (20)
where wno stands for winding number around the origin.
Since both L∗d(jω) and Li(jω, ρ) are constant or zero
for the semi-circle with infinity radius of the Nyquist
contour the wno depends only on the variation of s in
the imaginary axis. Thus,
wno{[1 + Ld(jω)]} = wno{[1 + Li(jω, ρ)]} (21)
Since Ld(jω) satisfies the Nyquist stability criterion
Li(jω, ρ) will do so and all zeros of 1 + Li(jω, ρ) will be in
the left-hand side of the complex plan. Since the zeros of 1 +Li(jω, ρ) are the closed-loop poles, the system will be
internally stable. 2
Remark I: The constraint in (15) is an inner convex
approximation of the non convex constraint in (14) or (10). The quality of this approximation depends on the choice of
Ld. It can be shown that better approximation is achieved
if Ld is close to Li(ρ) (Karimi and Galdos [2010]). Note
thatLi(ρ) can be written as:
Li(ρ) = C(ρ)Gn+C(ρ)(Pi− P0) (22)
Therefore,Ldcan be seen as the desired open loop transfer
function of the system without dead-time, i.e., C(ρ)Gn.
This will be a good approximation ifτiis close toτnor the
model mismatchPi− P0 is small. For systems with large
uncertainty in the time delay, for each modelPi different
Ld can be considered. In this case,Ldi can be computed
as follows:
Ldi=Ld+ L d
Gn
(Pi− P0) (23)
Note that Ldi should not encircle the critical point. An
iterative algorithm can be used to improve the results. In
this algorithmLd in the (k + 1)-th iteration is computed
using the controller of the last iterationCk(ρ) and (22):
Ldi=Ck(ρ)Gn+Ck(ρ)(Pi− P0) (24)
3.2 Primary controller design
The problem of minimizing the upper boundγ of the
infin-ity norm of the weighted sensitivinfin-ity function is considered. Therefore, the primary controller should be obtained from the following optimization problem:
min
ρ γ
Subject to: (25)
|W1Si|+|W2Ti|∞< γ for i = 1, . . . , q
This optimization can be convexified using Theorem 1 and solved by an iterative bisection algorithm. At each
iteration j, γj is fixed and W1 and W2 are replaced
by W1/γj and W2/γj. Then, a feasibility problem is
solved under the convex constraints (15). If the problem is feasible,γj+1is chosen smaller thanγj. Otherwiseγj+1 is
increased.
Notice that the condition (15) is defined for every
fre-quency ω leading to infinite number of constraints. In
practice, a frequency grid can be used with a sufficiently
large number of frequency points N (a finer grid can
be used around the crossover frequency). The effect of gridding on the stability and performance of the closed loop system has been studied in Galdos et al. [2010a]. Example 1 Consider the process described by (1) with multiplicative uncertainty as in (2) with
Gn(s) = 1 (5s + 1)(10s + 1) (26) and W2(s) = −s 2− 2s s2+ 2s + 1 (27)
The unknown time delayτ belongs to the set [4.5, 5, 5.5].
The nominal model used in the SP structure is chosen as P0(s) = Gn(s)e−5s. The performance specification is
defined by the following filter:
W1(s) = 2/(30s + 1)2 (28)
A PID primary controller withTf = 0.01 that minimizes
|W1Si|+|W2Ti|∞< γ for i = 1, 2, 3 should be computed.
Since the controller has an integrator, Ld is chosen as
Ld(s) = ωc/s where ωc = 0.1 rad/s which is 20%
problem (25) is solved considering N = 100 equally
spaced frequency points between 10−3and 103rad/s. The
resulting primary controller is:
C(s) = 12.3s2+ 3.28s + 0.2201
0.01s2+s (29)
and leads to γ = 0.313. This controller is compared to
that proposed in Kaya [2001]. Kaya’s controllers performs better than other controllers presented in the literature (Palmor and Blau [1994], Hagglund [1992] and Hang et al. [1995]). Fig. 3 depicts the performance of both controller on unitary step setpoint change considering the time-delay τ = 4.5s, τ = 5.0s and τ = 5.5s. As it can be seen, both controller performed well, however, the proposed controller achieves faster response.
0 5 10 15 20 25 30 35 40 0 0.5 1 Outputs 0 5 10 15 20 25 30 35 40 0 0.5 1 Outputs 0 5 10 15 20 25 30 35 40 0 0.5 1 Outputs Time [s] τ = 4.5s τ= 5.0s τ= 5.5s
Fig. 3. Example 1: Blue solid line: proposed; black dot-dashed line: ref Kaya [2001]
4. EXTENSION TO UNSTABLE SYSTEMS The SP in the scheme shown in Fig. 1 cannot be used for unstable plants since the controller will contain zeros in right-hand side of the s-plan which cancel the unstable poles in the plant and leads to instability. To avoid this unstable zero-pole cancellation, the control structure shown in Fig.1 should be changed. Several alternatives are available in literature to cope with unstable processes with dead-time (see, for example, De Paor [1985], Majhi and Atherton [1998], Liu et al. [2005], Normey-Rico and Camacho [2007, 2009]). Consider, for instance, the SP with modified dead-time free model depicted in Fig. 4 which is discussed in Normey-Rico and Camacho [2007]. In this
case, the dead-time free model is defined asGm=NDmn and
the equivalent controller becomes:
+ + + + − − r C(s) u P (s) Gm(s) P0(s) yp y
Fig. 4. Smith Predictor with modified dead-time free model Ceqm(ρ) = C(ρ) 1 +C(ρ)H (30) where H = Gm− P0= (Nm− Nne−τs) 1 Dn
Note that the poles ofH are zeros of Ceqm(ρ). Therefore,
Nm must be tuned such that the zeros of Nm− Nne−τs
cancel the unstable poles in Dn. Once Nm has been
properly designed, the primary controller can be obtained by solving the optimization problem in (25) redefining H = Gm− P0 andLi(ρ) = (Gm− P0+Pi)C(ρ).
Here, care should be taken in the choice ofLd. As it has
been shown, thewno of 1 + Li equals the wno of 1 + Ld.
Therefore, Ld should be chosen such that the number of
encirclement of the critical point (−1 + 0j) by its Nyquist
plot is equal to the number of unstable poles inPi.
Example 2 Consider the model studied in Meinsma and Zwart [2000] given by:
P (s) = s − ak [1 + ∆(s)W2(s)]e−τs (31)
where k = 1, a = 1, τ = τn ± 0.02 and τn = 0.2.
The interval of variation of τ is gridded using q = 3
equally spaced points. A finer grid just increases the number of constraints and for this example does not change significantly the final controller. The performance and uncertainty filters are respectively chosen as:
W1(s) = 210s + 1s + 1 and W2(s) = 0.2s + 1.1s + 1 (32)
Here, we use the SP with modified dead-time free model (Fig. 4) due to its simplicity. The dead-time free model Gm(s) is chosen as
Gm(s) =T ms + 1
s − 1 . (33)
Tmis computed in order to obtainH(s) = Gm(s) − P0(s)
without a pole ins = 1. Since
H(s) = s − 11 [Tms + 1 − e−0.2s], (34)
ifTm=e−0.2− 1, then s = 1 is a zero of H(s).
A PI as the primary controller is designed. The first step is
to choose the transfer functionLd(s), which must encircle
the critical point in the Nyquist diagram once and must contain one integrator. Therefore, it is chosen as
Ld(s) = 10
s + 1
s(s − 1). (35)
Optimization problem (25) is solved consideringN = 100
equally spaced frequency points between ω = 10−3rad/s
andω = 103rad/s and the following controller is obtained: C0(s) = (3.582s + 0.5838)/s (36)
which yields γ = 0.6854. This result can be further
improved by using a new Ld(s) based on C0(s) in the
optimization problem. With this newLd(s) = Gm(s)C0(s)
the optimal primary controller is:
C(s) = (2.994s + 0.4612)/s (37) andγ = 0.6074. Figure 5 depicts the function
Γi(jω) = |W1(jω)Si(jω)| + |W2(jω)Ti(jω)|
where Si and Ti are respectively given by (7) and (8)
Note that the maximum value of the function is 0.6072,
which occurs when τ = τn + 0.02 = 0.22, is close to
the bound γ. It is worth to point out that, although
the conditions given in Theorem 1 are only sufficient to
guarantee Γi∞ < γ, with a proper choice of Ld it is
possible to obtain a solution with very low conservatism. Furthermore, the resulting controller is a standard PI which can be implemented in a straightforward manner and has great practical significance.
10−2 100 102 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Frequency, [rad,s] Magnitude τn−∆τ τn+∆τ
Fig. 5. Example 2: Blue solid line: Γ for τ = 0.18; green
solid line: Γ for τ = 0.2; black solid line: Blue solid
line: Γ forτ = 0.22; red dot-dashed line: γ.
For the same example, controller designed in Meinsma and
Zwart [2000] leads to the optimal γ = 0.9407 which is
55% higher than the value obtained with the proposed method. It should be mentioned that the true robust performance criterion in (25) is not minimized in Meinsma and Zwart [2000]. Instead, the maximum singular value of [W1(jω)S(jω) W2(jω)T (jω)] for all ω is minimized by
theH∞ control theory.
5. GAIN-SCHEDULED CONTROLLER DESIGN
Consider an uncertain plantP (s, θ) belonging to the set:
P = {G(s, θ)e−τi(θ)s, i = 1, . . . , q} (38) where the dead-time free part of the model has unstruc-tured multiplicative uncertainty and is described as:
G(s, θ) = Gn(s, θ)[1 + ∆(s)W2(s)] (39)
and θ is a vector of scheduling parameters that belongs to a finite set Θ ={θ1, θ2, . . . , θm} (corresponding e.g. to
the different operating point parameters). It is assumed that the operating point does not frequently change (the stability and performance are achieved for the frozen scheduling parameter). The dead-time is also a function of the scheduling parameter and uncertain, so for a given value ofθ it belongs to the set {τ1(θ), τ2(θ), . . . , τq(θ)}.
We will consider the SP shown in Fig. 6 where both, the nominal model P0(s, θ) = Gn(s, θ)e−τn(θ)s and the
primary controllerC(s, θ) are functions of the scheduling
parameter vectorθ. The goal is to compute a primary
gain-scheduled controller for this scheme that meets the H∞
robust performance specification.
The primary controllerC(s, θ) is linearly parametrized by:
C(s, θ) = ρT(θ)φ(s), where the basis function vector φ(s)
is defined in (5) andρT(θ) is given by
+ + + + − − r C(s, θ) u P (s, θ) Gn(s, θ) e−τn(θ)s yp y
Fig. 6. Gain-Scheduled Smith Predictor ρT
(θ) = [ρ1(θ), ρ2(θ), . . . , ρn(θ)] (40)
Every gain is a polynomial function of order δ of the
scheduling parameters and is defined as
ρi(θ) = (νi,δ)Tθδ+. . . + (νi,1)Tθ + νi,0 (41)
andθk denotes element-by-element power ofk of vector θ.
The sensitivity and complementary sensitivity functions are respectively given by:
Si(s, θ) = 1 +C(s, θ)H(s, θ) 1 +C(s, θ)(H(s, θ) + Pi(s, θ)) (42) Ti(s, θ) = C(s, θ)P i(s, θ) 1 +C(s, θ)(H(s, θ) + Pi(s, θ)), ∀θ ∈ Θ
whereH(s, θ) = Gn(s, θ)−P0(s, θ). The primary controller
is obtained from the following optimization problem: min
ρ γ
Subject to: (43)
|W1Si(s, θ)| + |W2Ti(s, θ)|∞< γ
fori = 1, . . . , q, ∀θ ∈ Θ
Optimization problem (43) is again solved using an itera-tive bisection algorithm as previously presented. At each iteration, a feasibility problem is solved with the following convex constraints: |W1(jωk)[1 +C(jωk, θl)H(jωk, θl)|+ |W2(jωk)C(jωk, θl)P (jωk, θl)| |1 + Ld(jωk)|− Re{[1 + L∗ d(jωk)][1 +Li(jωk, θl)]} < 0 fork = 1, . . . , N, i = 1, . . . , q, l = 1, . . . , m (44) Example 3 The design method is applied on a simulated system having a resonance whose frequency changes as
a function of a scheduling parameter θ. Consider the
following plant model
P (s, θ) = G(s, θ)e−τs (45) whereG(s, θ) = Gn(s, θ)[1 + ∆(s)W2(s)] and Gn(s, θ) = (2 + 0.2θ)2 s2+ 0.2(2 + 0.2θ)s + (2 + 0.2θ)2 (46) W2(s) = 0.81.1337s 2+ 6.8857s + 9 (s + 1)(s + 10) (47)
and θ ∈ [−1, −0.5, 0, 0.5, 1]. Consider also that the
dead-time is within the interval τ ∈ [2.7, 3.0, 3.3] but its
ex-act value is unknown in runtime. The objective is to design a primary gain-scheduling PID controller for the Smith Predictor structure considering the performance filterW1(s) = (20s+1)2 2. The parametersρ of the primary controller will be affine functions of the scheduling param-eterθ. The filter of the derivative action is chosen to have a time constant ofTf = 0.01s.
Finally, optimization problem (43) is solved considering Ld = 1/s and N = 100 equally spaced frequency points
between 10−2 and 102rad/s. The resulting gain-scheduled
controller is given by:Kp(θ) = −0.0168θ+0.2152, Ki(θ) =
0.0144θ + 2.4736, Kd(θ) = −0.1224θ + 0.6424.
This controller leads to:
|W1Si(s, θl)| + |W2Ti(s, θl)|∞< γ = 0.8928 (48)
l = 1, . . . , 5, i = 1, 2, 3
The gain-scheduled controller is evaluated consideringθ =
−1, 0, 1 and τ = 3.3s. The performance is compared to a
fixed-gain PID designed for the nominal case (θ = 0 and
τ = 3s). Figure 7 shows the step response of the gain-scheduled controller in all conditions (blue, red and green solid lines) compared with the fixed PID controller (black dashed line, highly oscillating).
0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outputs Time [s]
Fig. 7. Example 3: Blue, red and green solid line:
gain-scheduled PID Smith Predictor and G2 using θ1 =
−1, θ3= 0 andθ5= 1 respectively; black dashed line:
fixed PID Smith Predictor usingθ = −1.
6. CONCLUSIONS
This paper presents a new method to design robust Smith predictor for uncertain time-delay systems using convex optimization techniques. The proposed approached allows one to design PI/PID as well as higher order primary controllers in the Smith predictor structure which provide
robustH∞ performance for systems with uncertain
dead-time and multiplicative or multimodel uncertainty in the dead-time free model of the system. The method is based
on a convex approximation of theH∞robust performance
criterion in the Nyquist diagram. This approximation relies on the choice of a desired open-loop transfer function
Ldfor the dead-time free model of the plant. The extension
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